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Let $K$ be a number field and $I$ be an ideal in the ring of integers $\mathcal{O}_K$. Let $n$ be the smallest positive integer in $I \cap \mathbb{Z}$. I'm trying to figure out what to call $n$.

When $I$ is prime $n$ is the characteristic of the residue field. Is it appropriate to call $n$ the "characteristic of $I$" for general $I$, or is this a horrid abuse of terminology? Is there something else that's widely accepted?

EDIT: It looks like ``characteristic'' can be defined for any ring, so $n$ is the characteristic of $\mathcal{O}_K/I$. Perhaps it's not too much of a stretch to call it the characteristic of $I$??

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This is just the norm of the ideal, no? –  Cam McLeman Sep 23 '10 at 16:50
    
@Cam: It's the norm of the ideal pulled back to $\mathbb{Z}$. –  Martin Brandenburg Sep 23 '10 at 17:00
    
No, it is not, but it divides the norm. –  felix Sep 23 '10 at 17:00
    
(That was a comment to Cam's question) –  felix Sep 23 '10 at 17:01
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I certainly wouldn't say "characteristic of $I$". "Residue characteristic of $I$" would be better. (This is what I, and many others, would say in the case when $I$ is a prime ideal, although using it in the non-prime case is unconventional, and would merit an explanation the first time you used it in your paper.) –  Emerton Sep 24 '10 at 3:19
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